Applying shrinkwrap modifier corrupts the shape
This is because the objects are Curve objects. Note the warning in the Info panel that appears when the modifier is applied:



When the modifier is active (not applied) it affects the polygon geometry generated by the curve object. When you apply the modifier it will only affect the Control Vertices (CV points) of the curve, giving a different result:



As you can see, the CV points are shrink-wrapped to the edges of the mesh, redefining the geometry defined by the curve.

To retain the shape, you must convert the object to a Mesh. This can be done with AltC> Convert > Mesh. This will take the geometry generated by the curve object and turn it into a mesh object:



Note that this will apply all modifiers as well. (See this post)